Today we will learn about Linear regression models:

- Introduction
- Equations
- Example

Graphical representation of Linear model

**Introduction**

In statistics, **linear regression** is an approach for modeling the relationship between dependent variable and

independent variable. The case of one independent variable is called simple linear regression. For more than one independent variable, the process is called multiple linear regression.

In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Such models are called linear models.

Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications. This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.

Linear regression has many practical uses. Most applications fall into one of the following two broad categories:

- Linear regression is used when to predict things, forecasting, or error reduction in every dataset to observed the values of X and Y.
- Given a variable
*y*and a number of variables*X*_{1}, …,*X*_{p}that may be related to*y*, linear regression analysis can be applied to quantify the strength of the relationship between*y*and the*X*_{j}, to assess which*X*_{j}may have no relationship with*y*at all, and to identify which subsets of the*X*_{j}contain redundant information about*y*.

Linear regression models are often fitted using the least squares approach. Conversely, the least squares approach can be used to fit models that are not linear models. Thus, although the terms **least squares** and **linear model** are closely linked, they are not synonymous.

**Equations**

Lets assume simple linear model, in simple linear model there is only one independent variable (X) and one dependent variable (Y).

Equation for Y on X

y= n∈+βx →1

Equation for X on Y

x= n∈+βy →2

Using equation 1 and 2 we will calculate the value of ∈ and β

taking summation of these equations

∑y= n∑∈+β∑x →3

∑x= n∑∈+β∑y →4

on multiplying eq (3) by x and eq (4) by y

∑xy= ∈∑x+β∑x² →5

∑xy= ∈∑y+β∑y² →6

By using these equations we will find the values of coefficient(β) and error(∈)

**Example**

x | x^2 | y | y^2 | xy | ||

69 | 4761 | 70 | 4900 | 4830 | ||

63 | 3969 | 65 | 4225 | 4095 | ||

66 | 4356 | 68 | 4624 | 4488 | ||

64 | 4096 | 65 | 4225 | 4160 | ||

67 | 4489 | 69 | 4761 | 4623 | ||

64 | 4096 | 66 | 4356 | 4224 | ||

70 | 4900 | 68 | 4624 | 4760 | ||

66 | 4356 | 65 | 4225 | 4290 | ||

68 | 4624 | 71 | 5041 | 4828 | ||

67 | 4489 | 67 | 4489 | 4489 | ||

65 | 4225 | 64 | 4096 | 4160 | ||

71 | 5041 | 72 | 5184 | 5112 | ||

total | 800 | 53402 | 810 | 54750 | 54059 |

800 = 12∈+810β →1

810 = 12∈+800β →2

54059 = 810∈+54750β →3

54059 = 800∈ +53402β →4

On solving these equations we will get the values of

∈= ?

β= ?

After getting the values we will compare the F-value, and p-value, According to these comparisions we will decide the relationship importance between the variables [which variable is dependent on which variable and how it will effect the model]

**Note**: In next article I will teach you how to make a linear regression model in R

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